Computing complexity measures of degenerate graphs

TL;DR

This paper introduces an efficient algorithm to compute the VC-dimension of degenerate graphs, leveraging a novel data structure, with applications in pattern counting and analysis of real-world networks.

Contribution

The paper presents a new algorithm for computing VC-dimension in degenerate graphs and demonstrates its practical effectiveness through implementations and experiments.

Findings

VC-dimension can be computed in time $n^{ ext{log } d+1} d^{O(d)}$

Efficient algorithms for counting bipartite patterns like bicliques and co-matchings

Insights into VC-dimension of real-world networks from large-scale experiments

Abstract

We show that the VC-dimension of a graph can be computed in time $n^{\log d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that see a specific subset of vertices inside of a (small) query set. The construction of this data structure takes time $O(d2^dn)$, afterwards queries can be computed efficiently using fast M\"obius inversion. This data structure turns out to be useful for a range of tasks, especially for finding bipartite patterns in degenerate graphs, and we outline an efficient algorithms for counting the number of times specific patterns occur in a graph. The largest factor in the running time of this algorithm is $O(n^c)$, where $c$ is a parameter of the pattern we call its left covering number. Concrete applications of this algorithm include counting the number of (non-induced) bicliques in linear time, the number of co-matchings in quadratic time, as well as a constant-factor approximation of the ladder index in linear time. Finally, we supplement our theoretical results with several implementations and run experiments on more than 200 real-world datasets -- the largest of which has 8 million edges -- where we obtain interesting insights into the VC-dimension of real-world networks.